All we know is that we should stay alert and not let our guards down.
I want to know if the bold part is grammatically correct. Do I have to say "and do not let our guards down" or "not let our guards down"?
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You do not have to say either of those.
This is a slightly shortened version of "and not to let our guard down." The verb form you may be looking for there is the "to let". It sounds a bit stilted when included in the sentence but is perfectly well understood.
The idiom is letting, or not letting our guard down, in the singular. These are not various troops out there doing service, it is our own sense of guarding ourselves, our own guard, sense of protection.